{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/rigorous-computation-of-solutions-of-semi","title":"Rigorous computation of solutions of semi-linear PDEs on unbounded domains via spectral methods","arxiv_id":"2302.12877","date":"2023-02-24","proceeding":null,"authors":["Matthieu Cadiot","Jean-Philippe Lessard","Jean-Christophe Nave"],"abstract":"In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\\subset H^{s}(\\mathbb{R}^{m})$ ($s\\geq1$) via computer-assisted proofs. Our approach is fully spectral and uses Fourier series to approximate functions in $H^{l}$ as well as bounded linear operators from $L^{2}$ to $H^{l}$. In particular, we construct approximate inverses of differential operators via Fourier series approximations. Combining this construction with a Newton-Kantorovich approach, we develop a numerical method to prove existence of strong solutions. To do so, we introduce a finite-dimensional trace theorem from which we build smooth functions with support on a hypercube. The method is then generalized to systems of PDEs with extra equations/parameters such as eigenvalue problems. As an application, we prove the existence of a traveling wave (soliton) in the Kawahara equation in $H^{4}(\\mathbb{R})$ as well as eigenpairs of the linearization about the soliton. These results allow us to prove the stability of the aforementioned traveling wave.","url_abs":"https://arxiv.org/abs/2302.12877v3","url_pdf":"https://arxiv.org/pdf/2302.12877v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"rigorous-computation-of-solutions-of-semi","repo_url":"https://github.com/matthieucadiot/proofkawahara.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"rigorous-computation-of-solutions-of-semi","repo_url":"https://github.com/matthieucadiot/localizedpatternsh","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"rigorous-computation-of-solutions-of-semi","repo_url":"https://github.com/matthieucadiot/localizedpatternsh.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}